Abstract
Electrons moving in a periodic potential experience a quantized energy spectrum, where the discrete energy bands are known as Bloch bands. In a magnetic field the spectrum further splits into highly degenerate Landau levels. The interplay between both effects leads to a complex fractal energy spectrum known as Hofstadter’s butterfly. This chapter provides an introduction into the theoretical description of the system in the absence of interactions in terms of magnetic translation symmetries. The topological properties of the system are further discussed in terms of topological invariants, the Chern numbers, which are directly related to the quantization of the Hall conductivity.
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Aidelsburger, M. (2016). Square Lattice with Magnetic Field. In: Artificial Gauge Fields with Ultracold Atoms in Optical Lattices. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-25829-4_2
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DOI: https://doi.org/10.1007/978-3-319-25829-4_2
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